1.) Make your own server via spare comp, bought up internet connection.
You might not need a terribly fast one if you intend to keep this strictly local, so DSL or Comcast (default subscriptions) should do.
2a.) Utilizing the trust low-tech solution of offline marketing, post flyers, pass by word of mouth in the community, etc. to those in your locale that you wish to know about your site.
2b.) Here's the catch: when you inform others of your site, tell them explicitly that your site will only be online from Hour A to Hour B (i.e. 8PM to 10PM).
They will require any of the following; login info that you provided for them, a keyword to say upon entering the chat room, a certain thread to post in on the BBS, etc.
Use your imagination with this one.
3.) Repeat & vary the process, keeping your local site online only when needed or desired.
For the days that you don't want to filter access, utilize all the traditional methods of blocking outsiders out.
Not a perfect solution, but it's another tool to use rather than just whitelisting (blacklisting is less effective).